Sequential Cross-Validated Bandwidth Selection Under Dependence and Anscombe-Type Extensions to Random Time Horizons

TL;DR

This paper develops a sequential, data-adaptive bandwidth selection method for change detection in time series, providing asymptotic theory under dependence, and extends to random time horizons with applications in finance and risk management.

Contribution

It introduces a novel sequential cross-validation approach for bandwidth selection in change detection, with theoretical guarantees under weak dependence assumptions and extensions to random stopping times.

Findings

Established asymptotic normality of the bandwidth selector under dependence.

Proved limit theorems for change-point detection with random stopping times.

Applicable to GARCH processes and stochastic monitoring scenarios.

Abstract

To detect changes in the mean of a time series, one may use previsible detection procedures based on nonparametric kernel prediction smoothers which cover various classic detection statistics as special cases. Bandwidth selection, particularly in a data-adaptive way, is a serious issue and not well studied for detection problems. To ensure data adaptation, we select the bandwidth by cross-validation, but in a sequential way leading to a functional estimation approach. This article provides the asymptotic theory for the method under fairly weak assumptions on the dependence structure of the error terms, which cover, e.g., GARCH($p,q$) processes, by establishing (sequential) functional central limit theorems for the cross-validation objective function and the associated bandwidth selector. It turns out that the proof can be based in a neat way on \cite{KurtzProtter1996}'s results on the weak convergence of \ito integrals and a diagonal argument. Our gradual change-point model covers multiple change-points in that it allows for a nonlinear regression function after the first change-point possibly with further jumps and Lipschitz continuous between those discontinuities. In applications, the time horizon where monitoring stops latest is often determined by a random experiment, e.g. a first-exit stopping time applied to a cumulated cost process or a risk measure, possibly stochastically dependent from the monitored time series. Thus, we also study that case and establish related limit theorems in the spirit of \citet{Anscombe1952}'s result. The result has various applications including statistical parameter estimation and monitoring financial investment strategies with risk-controlled early termination, which are briefly discussed.